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? (24)
L1 (x, ?, ? , . . . , ) = 0
1

on solutions of equation (22), such that equation (22) together with (24) is invariant
with respect to the set of operators Y . This means that the following conditions are
satisfied:
? ? ?
Y k L = R0 L + R1 L1 , Y k L = R2 L + R3 L1 ,
s s

where R0 , R1 , R2 , R3 are some smooth functions, Y k is the s-th Lie prolongation of
s
the operator Yk ? Y .
It is evident that Definition 1 makes sense only if system (23), (24) is compatible.
The notion of conditional symmetry has turned out extremely efficient, and during
recent years it was established that d’Alembert, Schr?dinger, Maxwell, heat, Boussi-
o
nesq equations possess nontrivial conditional symmetry. The problem of detailed
description of conditional symmetry for principal equations of mathematical physi-
cs remains open [6, 7].
Theorem 5 [2, 8]. Equation
p2
p0 ? a
(25)
? = F (|?|)?
2m
Galilei invariant nonlinear Schr?dinger type equations
o 361

is conditionally invariant with respect to the operator
? ?
+ ?? ? ? i ln(?(? ? )?1 )Q, (26)
Y = xa pa + r ?
?? ??
if
?1 ?1
+ a2 |?|?2r
F = a1 |?|2r (27)
, r=0

1
?
L1 (u) = ?|?| ? a3 |?|(r?2)/r = 0, r, a1 , a2 ? R. (28)
a3 = a2 m,
2
Corollary 4. Operator (26) generates the following finite transformations

x0 > x0 = x0 , xa > xa = exp ? · xa , (29)

? > ? = exp(r?) exp{exp(2?)}(?(? ? )?1 )1/2 |?|, (30)

? is the group parameter.
Formula (30) gives nonlinear transformations for the function ?.
So equation (25), (27) together with (28) admits an additional operator Y (26).
Equation (25) with the nonlinearity (27) without the additional condition (28) is not
invariant with respect to the operator (26).
Having the additional symmetry operator (26) we can construct new ansatzes.

5. Reduction and exact solutions of nonlinear equations
Let us consider the simplest equations (1), (2) which are invariant with respect to
algebra AG2 (1, 3):
??
= ??? + ?|?|4/3 ?, (31)
i
?t
?? ?|?| ?|?|
= ??? + ?|?|?2 (32)
i ?.
?t ?xk ?xk

We shall search for the solutions in the form [7]

w ? (w1 , w2 , w3 ), (33)
? = f (t, x)?(w), wk = wk (t, x).

Definition 2. We shall say that the formula (33) is an ansatz for equations (31), (32)
if functions f (x), w1 , w2 , w3 have such structure that four-dimensional equations are
reduced to three-dimensional ones for the function ?(w). Equations obtained for ?(w)
depend only on w.
The problem of reduction in the general formulation is an extremely difficult
problem and it requires explicit description of functions f (x), w1 , w2 , w3 which
satisfy a nonlinear system of equations. We do not think that it is possible now to
construct the general solution of these equations. But in case of an equation having
rich symmetry properties the problem of reduction and description of f (x) and w can
be partially reduced to an algebraic problem of description of inequivalent subalgebras
of this equation [7].
362 W.I. Fushchych

By means of subalgebraic structure of the algebra AG2 (1, 3) we have constructed
quite a large list of ansatzes which reduce four-dimensional equations (31), (32) to
three-dimensional ones. I adduce some of them.
Ansatzes for equations (31), (32).
x2
3
1. ?(x) = exp i ?(w),
4t (34)
x2
w3 = x3 ? t arctan .
x2 x2 ,
w1 = t, w2 = +
1 2
x1
The reduced equation
?2? 2
?2?
?? ? w3 ?? w1
= ?4w2 2 ? 1 + 2 4/3
(35)
i + + 2 + ?|?| ?.
?w1 2w1 w1 ?w3 ?w2 w2 ?w3
|x|2 t
i
? = (t2 + 1)?3/4 exp
2. + 2? arctan t ?(w),
1 + t2
4 (36)
x1 x2 x3
w1 = v w2 = v , w3 = v
, .
1 + t2 1 + t2 1 + t2
The reduced equation
(2? ? ww)
?2? ?2? ?2?
? 2? 2 ? ?w 2 ? ? + ?|?|4/3 ? = 0, (37)
?w1 ?w2 4
3
where ? is an arbitrary real parameter.
|x|2 t tx2 ? x1
i
?3/4
2
3. ? = (t + 1) exp + 2? 2 arctan t ?(w),
2
4
1+t t +1
(38)
tx1 + x2 tx2 + x1 x3
, w3 = v
w1 = 2 + ? arctan t, w2 = 2 .
t +1 t +1 t2 + 1
The reduced equation
?? ?? ?? 1
? w2 = ?? + (2?w2 + ww)? + ?|?|4/3 ?. (39)
i? + w1
?w1 ?w2 ?w1 4
Having investigated symmetry of reduced equations which depend on three vari-
ables and then of ones depending on two variables we come finally to ordinary di-
fferential equations of the form
d2 ? d?
+ C(w)? + ?|?|4/3 ? = 0, (40)
A(w) + B(w)
2
dw dw
where A(w), B(w), C(w) are second degree polynomials.
Having solved equations (40) we construct exact solutions of the four-dimensional
nonlinear Schr?dinger equations (31) by means of the formulae (34), (36), (38).
o
Solutions of equation (32) constructed by means of ansatzes (34), (36), (38).
exp(ia0 t)
?(t, x) = v , ? > 0, a0 < 0;
{ ?? cos(ax)}3/2
exp(ia0 t)
?(t, x) = v , ? > 0, a0 > 0;
{ ?? sh(ax)}3/2
exp(ia0 t)
?(t, x) = v , ? < 0, a0 > 0;
{ ?? ch(ax)}3/2
Galilei invariant nonlinear Schr?dinger type equations
o 363

ak are arbitrary real parameters and what is more aa = a2 = 4 |a0 |, ? = 3?/5a0 .
9
One can see that all obtained solutions depend non-analytically on the parameter ?
(constant of interaction).
The obtained three-dimensional partial solutions can be used for construction of
multi-parameter families of exact solutions. Really, as equation (31) is invariant with
respect to 13-parameter group G(1, 3), that means the following.
If ?1 (t, x) is a solution of equation (31), then functions
v 2t
i
?2 (t, x) = exp vx + ?1 (t, x + vt),
2 2
(41)
x ? vt
i ?x 2 + 2vx + v 2 t t
(1 ? ?t)?3/2 ?1
?3 (t, x) = exp ? ,
1 ? ?t 1 ? ?t 1 ? ?t
4
are also solutions of the same equation. v, ? are real parameters.

6. Galilei invariant nonlinear equations
with second order derivatives
Now we formulate one result about the equations (1 ) which are invariant under
AG2 (1, n) (for more details, see [9]).
Theorem 6 [9]. The equations
?? ? ?? ?
?? ??
S? = A0 ?? ? ? ?1 + (? ? )?1 ? ?? ? ? (? ? )?1 +
?xa ?xa ?xa ?xa
2n+4 ?|?| ?|?|
+ A1 |?|4/n ? + A2 |?|? n ?
?xa ?xb
? 2 ?? ?? ? ?? ?
?2? ?? ??
? ? ?1 + (? ? )?1 ? ? (? ? )?1
? ,
?xa ?xb ?xa ?xb ?xa ?xb ?xa ?xb
?|?| ?|?| ? 2n+4
A0 ? A0 (w), A1 ? A1 (w), A2 ? A2 (w) w = |?| n
?xa ?xa
are invariant under AG2 (1, n) algebra. A0 , A1 , A2 are arbitrary smooth functions.

7 Acknowledgments
This research was supported by Ukrainian Committee of Sciences and Technology.
I would like to thank Professor H.-D. Doebner for invitation to International Symposi-
um “Nonlinear, Deformed and Irreversible Quantum Systems”.

1. Fushchych W., Serov M., On some exact solutions of the three-dimensional nonlinear Schr?dinger
o
equation, J. Phys. A, 1987, 20, 16, L29–L33.
2. Fushchych W., Chopyk V., Symmetry and non-Lie reduction of the nonlinear Schr?dinger equation,
o
Ukr. Math. J., 1993, 45, 4, 581–597.
3. Schuh D., Chung K.-M., Hartman H.N., Nonlinear Schr?dinger-type field equation for the descrip-
o
tion of dissipative systems, J. Math. Phys., 1983, 24, 6, 1652–1660.
4. Briihl L., Lange H., The Schr?dinger–Langevin equation: special solutions and nonexistence of
o
solitary waves, J. Math. Phys., 1984, 25, 4, 786–790.
5. Fushchych W., Cherniha R., Galilei invariant nonlinear equation of Schr?dinger type and their exact
o
solutions: I, II, Ukr. Math. J., 1989, 41, 10, 1349–1357; 12, 1687–1694.
6. Fushchych W., Nikitin A., Symmetries of Maxwell’s equations, D. Reidel, 1987.
364 W.I. Fushchych

7. Fushchych W., Shtelen W., Serov M., Symmetry analysis and exact solutions of equations of nonli-
near mathematical physics, Kluwer Academic Publishers, 1993.
8. Fushchych W., Chopyk V., Conditional invariance of the nonlinear Schr?dinger equation, Proc.
o
Ukrainian Acad. Sci., 1990, 4, 30–33.
9. Fushchych W., Cherniha R., Systems of nonlinear evolution equations of the second order invariant
under the Galilean algebra and its extensions, Proc. Ukrainian Acad. Sci., 1993, 8, 44–51.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 365–370.

On new exact solutions
of the multidimensional nonlinear
d’Alembert equation
W.I. FUSHCHYCH, A.F. BARANNYK, YU.D. MOSKALENKO

On the present paper new classes of exact solutions of the nonlinear d’Alembert
equation in the space R1,n , n ? 2,

2u + ?uk = 0 (1)
2
are built. Here 2u = u00 ?u11 ?· · ·?unn , uµ? = ?xµ ?x? , u = u(x), x = (x0 , x1 , . . . , xn );
?u

µ, ? = 0, 1, . . . , n. Symmetry properties of equation (1) have been studied in papers
[1, 2] in which it was established that equation (1) is invariant under the extended
?
Poincar? algebra AP (1, n):
e

J0a = x0 ?a + xa ?0 , Jab = xb ?a ? xa ?b , Pµ = ?µ ,
2u
S = ?xµ ?µ + ?u (a, b = 1, . . . , n; µ = 0, 1, . . . , n).
k?1
?

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